Compute $17^9 \div 17^7$.
We write $17^9=17^2\cdot 17^7$ using the product-of-powers property, which says that $a^{m+n} = a^ma^n$. We get \[
17^9 \div 17^7 = 17^2\cdot 17^7 \div 17^7 = 17^2 \cdot 1 = \boxed{289},
\]since any nonzero number divided by itself equals 1.